Edge Effect ... A New Wrinkle

by Kim Iles (who wrote this article) and Jim Flewelling (who asked the right question)

John has a frequent column called "It's the people". Well it is. A short time ago Jim Flewelling and I were visiting in my office. Jim was saying he liked my new "toss-back" method, but had a surprising question "Kim, why do you bother with volume/acre - why not just get total volume?"

"Simple question," you might think. Never thought of it, myself. We just routinely think we should get the volume/acre and multiply by acres. We don't need to do that, it is just a habit. In this case, a habit that caused a blind spot. However, as soon as we looked at it that way, the toss-back method was easier to prove, and the mild restriction that you need "at least one point in the area" drops away. A small, but significant improvement.

Let me outline this view of the situation, and show you how it would work even if you did not have a single point inside the area (which might be the case with riparian zones).

Let's work with a stand of "x" acres. The area is secret for the moment. We will use a grid of sample points in a large square, 1 point per acre. The total area covered by the square (which is much larger than the stand) is 100 acres. It could be any sized area. It will not matter as long as it covers much more than the stand itself. With this large area there are no trees in the stand that would be "in" with a prism outside that large area. Put another way, all the circles surrounding trees in the stand are inside the large area.

To place the grid of 1 point/acre just pick a random point in the lower left acre, and this will be the start of the grid. The grid extends from that point over the whole area, and will always give you exactly 100 points. This is neat, because the sample size is now always the same, and a variable sample size would cause nasty logical complications.

We want to get the total basal area of ALL the trees on this stand. We take a count at every one of the 100 points. Most of them will be zero. The average TC on all 100 points times the BAF times total area will be the total basal area inside the large square. This is clearly an unbiased total, and this is easy to prove. That's it. We are done.
A BAF of 20 and a total count of 50, gives a total basal area of (20 BAF * 50 trees / 100 points * 100 acres) = 1,000 square feet on the total area surrounding the stand (and therefore on the stand itself). Using algebra, you can shift the number of points under the number of acres, and get exactly the same result with this simpler equation :

BA = (BAF * TC * Acres per point)

With 200 points in the same area (therefore giving a count of 100), that would be (20*100*.5 acres/point = 1,000 square feet). The same answer. What is the basal area per acre on the stand? We do not know. Do you care?

If you have measured some VBAR trees you could also calculate the total volume. You do not know the volume per acre, but you do know the total.

Now here is something rather neat. We do not need a traverse of the area. We do not even need to know where the precise boundary was. All we need to know is whether the trees we are looking at are inside the stand. We do not even need to know if WE are in the stand (a slightly relaxed requirement from the original toss-back method reported earlier, where we did need to know if we were in the stand).

Do you want to know the volume/acre? Well, just divide the total volume by the area (which you would now need to figure out). Looking at the process in this way, we can derive the toss-back method we explained in the earlier issue of the newsletter, but from a slightly different logic.

It will not change the total if we shift any of the "in" trees from some of the points to other points. The total tree count, basal area and volume will remain exactly the same. So let's do it. Let's select some points in OR NEAR the stand, and "toss back" any data from the outer points onto these points. Our total will not change at all, and if we know the area, the volume/acre will not change, yet we have eliminated any potential bias from edge effect. It has been taken care of by the counts in the outer area.

If the grid is pretty tight, we can choose the points actually inside the stand to accumulate the data onto. If we have a long thin stand, where we might not even have one point in the stand, we can accumulate the data on several points in the nearby area of the stand. The only reason we care to do this shifting is that we want to calculate statistics on the volume or basal area, and this sort of accumulation really helps give a more reasonable approximation (it is not exact because it is a systematic sample).

Statistics would be done on the points that accumulate the data, just as you would do it with current methods. If you know the average basal area on the points absorbing the data within 6%, then you will know the total basal area (and therefore the basal area/acre) within 6% as well. So, as it stands, we have a pretty simple method.

  1. Put in a standard grid, making sure that it more than covers the stand and the plot spacing is correct. Choose some of the points to accumulate data onto (before you examine the results).

  2. Calculate the total basal area or volume.

  3. If you need to know the volume/acre, find the stand area and divide it into the total.

  4. Calculate the statistics based on the points that are accumulating data, whether they are inside the stand or not.

The greater generality of this method, and the simplicity of the proofs, is due to asking an unusual question - "why not just calculate the total, and forget the acres?"

The wonderful thing about mixing people (especially curious people) is that they think in different ways, and that causes them to ask different questions, look for different viewpoints, and interpret equations with different logic. In the end, it's the people that cause so much advancement and joy in the world. Full credit to Jim Flewelling for his insight and improvement of the method (which also seems to be roughly the same as an earlier Japanese author who did not apparently realize the implications)

Originally published July 2002

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